In [1]}, it has been proven that \(\mathbf {A}\)  is self-adjoint and dissipative. Since \(\mathbb {H}^{2} \hookrightarrow H^{2}(\Omega ) \times H^{2}(\Gamma )\)  is continuous, then \(\mathbb {H}^{2} \hookrightarrow \mathbb {L}^2\)  is compact. By [2]}, we deduce that \(\mathbf {A}\)  has compact resolvent. Therefore, the operator \(\mathbf {A}\)  satisfies Assumption I.
